I read on wikipedia theat the upper convected time derivatvie is the rate of change of some tensor property of a small parcelof fluid that is written in the coordinate system rotating and stretching with the fluid. But concretely how do we obtain its expression ?
if you want more details on how this derivation works, this work is a great resource. Look specifically at Page 23 onwards. If you have any specific question about the derivation, feel free to ask me and I can go through some of it for you.
Hope that helps!